Determine the mean value of the loss immediately before the win when applying the doubling strategy, i.e., determine \(E\left(X_{N-1}\right)\) (example 10.6).

Data from Example 10.6

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Example 10.6 The structure of X given by (10.14) includes as a special case the winnings (losses) development when applying the doubling strategy: Jean bets 1 on the first game. If he wins, he gets 1. If he loses, his 'winning' are - 1. Hoping to equalize the loss, Jean will bet 2 on the next game. If he wins, he will get 2 and, hence, will have made total winnings of 1. But if he loses he will have total 'winn- ings' of -3, and will bet 4 on the next game and so on. After the first win Jean stops gambling. The following table shows the losses (winnings) development of Jean if he loses 5 times in a row and then wins: game 1 2 3 4 5 6 result loss loss loss loss loss win bet 1 2 4 8 16 32 'total winnings' -1 -3 -7 -15 -31 +1